Is it true that \[\sum_{n\in A}\frac{1}{n}<\infty?\]

OPEN

Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$ and $b,c>a$. Is there such an $A$ with
\[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>0?\]
Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with
\[\lvert A\cap\{1,\ldots,N\}\rvert<N^{1-c}?\]

Is it true that \[\sum_{n\in A}\frac{1}{n}<\infty?\]

Asked by Erdős and Sárközy [ErSa70], who proved that $A$ must have density $0$. They also prove that this is essentially best possible, in that given any function $f(x)\to \infty$ as $x\to \infty$ there exists a set $A$ with this property and infinitely many $N$ such that
\[\lvert A\cap\{1,\ldots,N\}\rvert>\frac{N}{f(N)}.\]
(Their example is given by all integers in $(y_i,\frac{3}{2}y_i)$ congruent to $1$ modulo $(2y_{i-1})!$, where $y_i$ is some sufficiently quickly growing sequence.)

An example of an $A$ with this property where \[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}\log N>0\] is given by the set of $p^2$, where $p\equiv 3\pmod{4}$ is prime.

For the finite version see [13].

SOLVED - $100

Let $A\subseteq \{1,\ldots,N\}$ be such that there are no $a,b,c\in A$ such that $a\mid(b+c)$ and $a<\min(b,c)$. Is it true that $\lvert A\rvert\leq N/3+O(1)$?

Asked by Erdős and Sárközy, who observed that $(2N/3,N]\cap \mathbb{N}$ is such a set. The answer is yes, as proved by Bedert [Be23].

For the infinite version see [12].

OPEN

Can every triangle-free graph on $5n$ vertices be made bipartite by deleting at most $n^2$ edges?

The blow-up of $C_5$ shows that this would be the best possible. The best known bound is due to Balogh, Clemen, and Lidicky [BCL21], who proved that deleting at most $1.064n^2$ edges suffices.

SOLVED

Does every triangle-free graph on $5n$ vertices contain at most $n^5$ copies of $C_5$?

Győri proved this with $1.03n^5$, which has been improved by Füredi. The answer is yes, as proved independently by Grzesik [Gr12] and Hatami, Hladky, Král, Norine, and Razborov [HHKNR13].

In [Er97f] Erdős asks more generally: if $r\geq 5$ is odd and a graph has $rn$ vertices and the smallest odd cycle has size $r$ then is the number of cycles of size $r$ at most $n^{r}$?

SOLVED

If $G$ is a graph with infinite chromatic number and $a_1<a_2<\cdots $ are lengths of the odd cycles of $G$ then $\sum \frac{1}{a_i}=\infty$.

OPEN - $1000

Does every graph with minimum degree at least 3 contain a cycle of length $2^k$ for some $k\geq 2$?

Conjectured by Erdős and Gyárfás, who believed the answer must be negative, and in fact for every $r$ there must be a graph of minimum degree at least $r$ without a cycle of length $2^k$ for any $k\geq 2$.

This was solved in the affirmative if the minimum degree is larger than some absolute constant by Liu and Montgomery [LiMo20] (therefore disproving the above stronger conjecture of Erdős and Gyárfás). Liu and Montgomery prove a much stronger result: if the average degree of $G$ is sufficiently large then there is some large integer $\ell$ such that for every even integer $m\in [(\log \ell)^8,\ell]$, $G$ contains a cycle of length $m$.

SOLVED

Is it true that for every infinite arithmetic progression $P$ which contains even numbers there is some constant $c=c(P)$ such that every graph with average degree at least $c$ contains a cycle whose length is in $P$?

SOLVED - $100

Is there a set $A\subset \mathbb{N}$ of density $0$ and a constant $c>0$ such that every graph on sufficiently many vertices with average degree $\geq c$ contains a cycle whose length is in $A$?

Bollobás [Bo77] proved that such a $c$ does exist if $A$ is an infinite arithmetic progression containing even numbers (see [71]).

Erdős was 'almost certain' that if $A$ is the set of powers of $2$ then no such $c$ exists (although he conjectured that $n$ vertices and average degree $\gg (\log n)^{C}$ suffices for some $C=O(1)$). If $A$ is the set of squares (or the set of $p\pm 1$ for $p$ prime) then he had no guess.

Solved by Verstraëte [Ve05], who gave a non-constructive proof that such a set $A$ exists.

Liu and Montgomery [LiMo20] proved that in fact this is true when $A$ is the set of powers of $2$ (more generally any set of even numbers which doesn't grow too quickly) - in particular this contradicts the previous belief of Erdős.

OPEN - $500

Let $f(n)\to \infty$ (possibly very slowly). Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $f(n)$ edges?

Conjectured by Erdős, Hajnal, and Szemerédi [EHS82].

Rödl [Ro82] has proved this for hypergraphs, and also proved there is such a graph (with chromatic number $\aleph_0$) if $f(n)=\epsilon n$ for any fixed constant $\epsilon>0$.

It is open even for $f(n)=\sqrt{n}$. Erdős offered \$500 for a proof but only \$250 for a counterexample. This fails (even with $f(n)\gg n$) if the graph has chromatic number $\aleph_1$ (see [111]).

OPEN - $500

Does every set of $n$ distinct points in $\mathbb{R}^2$ determine $\gg n/\sqrt{\log n}$ many distinct distances?

A $\sqrt{n}\times\sqrt{n}$ integer grid shows that this would be the best possible. Nearly solved by Guth and Katz [GuKa15] who proved that there are always $\gg n/\log n$ many distinct distances.

A stronger form (see [604]) may be true: is there a single point which determines $\gg n/\sqrt{\log n}$ distinct distances, or even $\gg n$ many such points, or even that this is true averaged over all points.

See also [661].

SOLVED

Let $f(m)$ be maximal such that every graph with $m$ edges must contain a bipartite graph with
\[\geq \frac{m}{2}+\frac{\sqrt{8m+1}-1}{8}+f(m)\]
edges. Is there an infinite sequence of $m_i$ such that $f(m_i)\to \infty$?

Conjectured by Erdős, Kohayakava, and Gyárfás. Edwards [Ed73] proved that $f(m)\geq 0$ always. Note that $f(\binom{n}{2})= 0$, taking $K_n$. Solved by Alon [Al96], who showed $f(n^2/2)\gg n^{1/2}$, and also showed that $f(m)\ll m^{1/4}$ for all $m$. The best possible constant in $f(m)\leq Cm^{1/4}$ is unknown.

OPEN - $250

Let $G$ be a graph with $n$ vertices such that every subgraph on $\geq n/2$ vertices has more than $n^2/50$ edges. Must $G$ contain a triangle?

A problem of Erdős and Rousseau. The constant $50$ would be best possible as witnessed by a blow-up of $C_5$ or the Petersen graph. Krivelevich [Kr95] has proved this with $n/2$ replaced by $3n/5$ (and $50$ replaced by $25$).

Keevash and Sudakov [KeSu06] have proved this under the additional assumption that $G$ has at most $n^2/12$ edges.

OPEN

Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy of $K_k$ in at least one of the $r$ colours. Prove that there is a constant $C=C(r)>1$ such that
\[R(n;3,r) < C^{\sqrt{n}}.\]

Conjectured by Erdős and Gyárfás, who proved the existence of some $C>1$ such that $R(n;3,r)>C^{\sqrt{n}}$. Note that when $r=k=2$ we recover the classic Ramsey numbers. Erdős thought it likely that for all $r,k\geq 2$ there exists some $C_1,C_2>1$ (depending only on $r$) such that
\[ C_1^{n^{1/k-1}}< R(n;k,r) < C_2^{n^{1/k-1}}.\]
Antonio Girao has pointed out that this problem as written is easily disproved, and indeed $R(n;3,2) \geq C^{n}$:

The obvious probabilistic construction (randomly colour the edges red/blue independently uniformly at random) yields a 2-colouring of the edges of $K_N$ such every set on $n$ vertices contains a red triangle and a blue triangle (using that every set of $n$ vertices contains $\gg n^2$ edge-disjoint triangles), provided $N \leq C^n$ for some absolute constant $C>1$. This implies $R(n;3,2) \geq C^{n}$, contradicting the conjecture.

Perhaps Erdős had a different problem in mind, but it is not clear what that might be. It would presumably be one where the natural probabilistic argument would deliver a bound like $C^{\sqrt{n}}$ as Erdős and Gyárfás claim to have achieved via the probabilistic method.

OPEN

Let $A\subset\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertices the points in $A$, where two vertices are joined by an edge if and only if they are an integer distance apart.

How large can the chromatic number and clique number of this graph be? In particular, can the chromatic number be infinite?

Asked by Andrásfai and Erdős. Erdős [Er97b] also asked where such a graph could contain an infinite complete graph, but this is impossible by an earlier result of Anning and Erdős [AnEr45].

See also [213].

OPEN

Let $\epsilon>0$ and $N$ be sufficiently large depending on $\epsilon$. Is there $A\subseteq\{1,\ldots,N\}$ such that no $a\in A$ divides the sum of any distinct elements of $A\backslash\{a\}$ and $\lvert A\rvert\gg N^{1/2-\epsilon}$?

It is easy to see that we must have $\lvert A\rvert \ll N^{1/2}$. Csaba has constructed such an $A$ with $\lvert A\rvert \gg N^{1/5}$.

OPEN - $100

Let $A\subset \mathbb{R}^2$ be a set of $n$ points. Must there be two distances which occur at least once but between at most $n$ pairs of points? Must the number of such distances $\to \infty$ as $n\to \infty$?

Asked by Erdős and Pach. Hopf and Pannowitz [HoPa34] proved that the largest distance between points of $A$ can occur at most $n$ times, but it is unknown whether a second such distance must occur.

It may be true that there are at least $n^{1-o(1)}$ many such distances. In [Er97e] Erdős offers \$100 for 'any nontrivial result'.

SOLVED

Let $f(n)$ be minimal such that every triangle-free graph $G$ with $n$ vertices and diameter $2$ contains a vertex with degree $\geq f(n)$.

What is the order of growth of $f(n)$? Does $f(n)/\sqrt{n}\to \infty$?

Asked by Erdős and Pach. The lower bound $f(n)\geq (1-o(1))\sqrt{n}$ follows from the fact that a graph with maximum degree $d$ and diameter $2$ has at most $1+d+d(d-1)=d^2+1$ many vertices.

Simonovits observed that the subsets of $[3m-1]$ of size $m$, two sets joined by edge if and only if they are disjoint, forms a triangle-free graph of diameter $2$ which is regular of degree $\binom{2m-1}{m}$. This construction proves that \[f(n) \leq n^{(1+o(1))\frac{2}{3H(1/3)}}=n^{0.7182\cdots},\] where $H(x)$ is the binary entropy function. In [Er97b] Erdős encouraged the reader to try and find a better construction.

In this note Alon provides a simple construction that proves $f(n) \ll \sqrt{n\log n}$: take a triangle-free graph with independence number $\ll \sqrt{n\log n}$ (the existence of which is the lower bound in [165]) and add edges until it has diameter $2$; the neighbourhood of any set is an independent set and hence the maximum degree is still $\ll \sqrt{n\log n}$.

Hanson and Seyffarth [HaSe84] proved that $f(n)\leq (\sqrt{2}+o(1))\sqrt{n}$ using a Cayley graph on $\mathbb{Z}/n\mathbb{Z}$, with the generating set given by some symmetric complete sum-free set of size $\sim \sqrt{n}$. An alternative construction of such a complete sum-free set was given by Haviv and Levy [HaLe18].

Füredi and Seress [FuSe94] proved that $f(n)\leq (\frac{2}{\sqrt{3}}+o(1))\sqrt{n}$.

The precise asymptotics of $f(n)$ are unknown; Alon believes that the truth is $f(n)\sim \sqrt{n}$.

SOLVED

Let $\epsilon,\delta>0$ and $n$ be sufficiently large in terms of $\epsilon$ and $\delta$. Let $G$ be a triangle-free graph on $n$ vertices with maximum degree $<n^{1/2-\epsilon}$.

Can $G$ be made into a triangle-free graph with diameter $2$ by adding at most $\delta n^2$ edges?

Asked by Erdős and Gyárfás, who proved that this is the case when $G$ has maximum degree $\ll \log n/\log\log n$. A construction of Simonovits shows that this conjecture is false if we just have maximum degree $\leq Cn^{1/2}$, for some large enough $C$.

In this note Alon solves this problem in a strong form, in particular proving that a triangle-free graph on $n$ vertices with maximum degree $<n^{1/2-\epsilon}$ can be made into a triangle-free graph with diameter $2$ by adding at most $O(n^{2-\epsilon})$ edges.

See also [618].

OPEN - $250

Let $A\subset \mathbb{R}^2$ be a set of $n$ points such that any subset of size $4$ determines at least $5$ distinct distances. Must $A$ determine $\gg n^2$ many distances?

Erdős also makes the even stronger conjecture that $A$ must contain $\gg n$ many points such that all pairwise distances are distinct.

SOLVED

Let $f(n)$ be the smallest number of colours required to colour the edges of $K_n$ such that every $K_4$ contains at least 5 colours. Determine the size of $f(n)$.

Asked by Erdős and Gyárfás, who proved that
\[\frac{5}{6}(n-1) < f(n)<n,\]
and that $f(9)=8$. Erdős believed the upper bound is closer to the truth. In fact the lower bound is: Bennett, Cushman, Dudek,and Pralat [BCDP22] have shown that
\[f(n) \sim \frac{5}{6}n.\]
Joos and Mubayi [JoMu22] have found a shorter proof of this.

OPEN - $500

Given $n$ distinct points $A\subset\mathbb{R}^2$ must there be a point $x\in A$ such that
\[\#\{ d(x,y) : y \in A\} \gg n^{1-o(1)}?\]
Or even $\gg n/\sqrt{\log n}$?

The pinned distance problem, a stronger form of [89]. The example of an integer grid show that $n/\sqrt{\log n}$ would be best possible.

It may be true that there are $\gg n$ many such points, or that this is true on average. In [Er97e] Erdős offers \$500 for a solution to this problem, but it is unclear whether he intended this for proving the existence of a single such point or for $\gg n$ many such points.

In [Er97e] Erdős wrote that he initially 'overconjectured' and thought that the answer to this problem is the same as for the number of distinct distances between all pairs (see [89]), but this was disproved by Harborth. It could be true that the answers are the same up to an additive factor of $n^{o(1)}$.

The best known bound is \[\gg n^{c-o(1)},\] due to Katz and Tardos [KaTa04], where \[c=\frac{48-14e}{55-16e}=0.864137\cdots.\]

SOLVED

Let $A\subset \mathbb{R}^2$ be a set of $n$ points. Can there be $\gg n$ many distinct distances each of which occurs for more than $n$ many pairs from $A$?

Asked by Erdős and Pach. Hopf and Pannowitz [HoPa34] proved that the largest distance between points of $A$ can occur at most $n$ times, but it is unknown whether a second such distance must occur (see [132]).

The answer is yes: Bhowmick [Bh24] constructs a set of $n$ points in $\mathbb{R}^2$ such that $\lfloor\frac{n}{4}\rfloor$ distances occur at least $n+1$ times. More generally, they construct, for any $m$ and large $n$, a set of $n$ points such that $\lfloor \frac{n}{2(m+1)}\rfloor$ distances occur at least $n+m$ times.

OPEN

Let $A\subset \mathbb{R}_{>0}$ be a set of size $n$ such that every subset $B\subseteq A$ with $\lvert B\rvert =4$ has $\lvert B-B\rvert\geq 11$. Find the best constant $c>0$ such that $A$ must always contain a Sidon set of size $\geq cn$.

For comparison, note that if $B$ were a Sidon set then $\lvert B-B\rvert=13$, so this condition is saying that at most one difference is 'missing' from $B-B$. Equivalently, one can view $A$ as a set such that every four points determine at least five distinct distances, and ask for a subset with all distances distinct.

Erdős and Sós proved that $c\geq 1/2$. Gyárfás and Lehel [GyLe95] proved \[\frac{1}{2}<c<\frac{3}{5}.\] (The example proving the upper bound is the set of the first $n$ Fibonacci numbers.)